Chapter 15 An Introduction to Differential Equations

15.1 🧟 Modeling a Zombie Outbreak in This Lecture Room

15.2 Objective

By the end of this activity, you should be able to:

  • Translate a real-world situation into a rate equation
  • Explain what a differential equation represents
  • Solve a simple differential equation using separation
  • Interpret a logistic-type solution
  • Understand why we start simple and then layer complexity

15.3 Part 1 — Start With the Messy Problem

There are 149 people in this lecture room.

At time \(t = 0\), one person becomes a zombie.

No further information is given.

15.3.1 Brainstorm Reality (Small Groups)

List at least 6 factors that might influence what happens next.










15.3.2 Reflection

If we tried to include all of these in a mathematical model:

  • Would it be solvable?
  • Would it be understandable?
  • Would we know how to measure everything?

Discuss briefly.

15.4 Part 2 — The Art of Simplifying

Modeling requires simplifying assumptions.

15.4.1 Create Simplifying Assumptions

In your group, create assumptions that make the problem as simple as possible while keeping the core mechanism.

Write your final list.










15.4.2 Identify the Core Mechanism

Complete:

Zombies increase because __________________________________________.

Focus on the single most important driver of change.

15.5 Part 3 — Translating Into Mathematics

Let:

  • \(S(t)\) = number of susceptible humans
  • \(Z(t)\) = number of zombies

We want to describe:

\[ \frac{dZ}{dt} \]






This means:

The rate at which zombies increase.

15.5.1 What Determines the Rate?

Discuss:

  • If there are very few zombies, what happens?
  • If there are very few humans, what happens?
  • Does growth depend on one group, or both?

Complete:

The rate of increase of zombies is proportional to ________________________.






















\[ \frac{dZ}{dt} = \beta ZS \]

where \(\beta > 0\).

15.5.2 Interpret the Structure

  • What does multiplying \(Z\) and \(S\) assume?
  • What real-world behavior does this encode?
  • What are the units of \(\beta\)?











15.6 Part 4 — Reducing the Model

Since no one enters or leaves:

\[ S + Z = 149 \]

15.7 Substitute

Solve for \(S\):

\[ S = 149 - Z \]

Substitute into the differential equation:

\[ \frac{dZ}{dt} = \beta Z(149 - Z) \]

Now we have one equation involving only \(Z\).

15.8 Part 5 — From Rate Rule to Function \(Z(t)\)

We now want to understand how \(Z\) changes over time.

15.8.1 Separate Variables

\[ \frac{dZ}{dt} = \beta Z(149 - Z) \]
















Separate:

\[ \frac{1}{Z(149 - Z)}\, dZ = \beta\, dt \]

15.8.2 Integrate Both Sides

\[ \int \frac{1}{Z(149 - Z)}\, dZ = \int \beta\, dt \]

Right side:

\[ \beta t + C \]

Left side (after partial fractions):

\[ \ln\!\left(\frac{Z}{149 - Z}\right) = 149\beta t + C \]

15.9 Part 6 — Isolate \(Z\)

We now solve for \(Z(t)\).

15.9.1 Step 1 — Exponentiate

\[ \frac{Z}{149 - Z} = Ce^{149\beta t} \]

15.9.2 Step 2 — Multiply Both Sides

\[ Z = Ce^{149\beta t}(149 - Z) \]

15.9.3 Step 3 — Distribute

\[ Z = 149Ce^{149\beta t} - Ce^{149\beta t}Z \]

15.9.4 Step 4 — Gather Terms

\[ Z(1 + Ce^{149\beta t}) = 149Ce^{149\beta t} \]

15.9.5 Step 5 — Solve

\[ Z(t) = \frac{149Ce^{149\beta t}} {1 + Ce^{149\beta t}} \]

Rewrite in standard logistic form:

\[ Z(t) = \frac{149} {1 + Ae^{-149\beta t}} \]

15.10 Part 7 — Interpret the Solution

Discuss:

  • What happens as \(t \to \infty\)?
  • What happens as \(t \to 0\)?
  • When is growth fastest?
  • What determines steepness?
  • How does the initial number of zombies affect the curve?

Sketch the curve and label:

  • Early slow growth
  • Rapid middle growth
  • Saturation near 149

15.11 Part 8 — Layering Complexity Back In

Our model assumed:

  • No one fights back.
  • Zombies never die.
  • Infection is instant.
  • The room is perfectly mixed.

Now we relax those assumptions.

15.11.1 Layer 1 — Humans Fight Back

\[ \frac{dZ}{dt} = \beta ZS - \delta Z \]

Rewrite:

\[ \frac{dZ}{dt} = Z(\beta S - \delta) \]

Zombies grow if:

\[ S > \frac{\delta}{\beta} \]

Discuss:

  • What does this threshold mean?
  • Could zombies fail to spread?

15.11.2 Layer 2 — Zombie Decay

\[ \frac{dZ}{dt} = \beta ZS - \delta Z - \mu Z \]

What happens if \(\mu\) increases?

15.11.3 Layer 3 — Human Reproduction

\[ \frac{dS}{dt} = rS - \beta ZS \]

\[ \frac{dZ}{dt} = \beta ZS - \delta Z \]

Before solving, predict:

  • Could coexistence occur?
  • Could oscillations occur?
  • What determines long-term behavior?

15.12 Final Reflection

Answer individually:

  1. Why did we start with an oversimplified model?
  2. What insight did the simple model give us?
  3. What changed when we added realism?
  4. What is a differential equation?

Complete:

A differential equation is a mathematical statement that describes __________________________________________.

15.13 The Modeling Workflow

You just followed the core structure of applied mathematics:

  1. Observe complexity.
  2. Strip to the core mechanism.
  3. Write a rate rule.
  4. Integrate to understand behavior.
  5. Add complexity deliberately.
  6. Reanalyze.

Differential equations are structured expressions of assumptions about change.